Let $A$ be a compact and connected subset of $\Omega$, where $\Omega$ is a open subset of $\mathbb{R}^{n}$. I would like to know if there exists a $\varepsilon>0$ such that $B=\{x \in A: d(x,\partial A)\geq \varepsilon\}$ is also a connected subset of $\Omega$ that is not empty.
I was trying to write set $B$ as the image of a continuous application by a connected set. I could not see a counterexample to this affirmation.
If any $\epsilon$ is ok, then take $\epsilon$ to be large enough so that your subset is $A$ (it exists because $A$ is compact) so $B$ is $A$ and thus connected. If you want to know if this is true for a certain sequence of epsilon of limit $0$, then $[0;1]$ in $\mathbb{R}$ will give you a counterexample (it won't work for $\epsilon < \dfrac{1}{2}$).